We present a survey of the fundamentals and the
applications of sparse grids, with a focus on the solution
of partial differential equations (PDEs). The sparse grid
approach, introduced in Zenger (1991), is based on a
higher-dimensional multiscale basis, which is derived from
a one-dimensional multiscale basis by a tensor product
construction. Discretizations on sparse grids involve O(N
(logN)d-1) degrees of freedom only, where d denotes
the underlying problem's dimensionality and where N is
the number of grid points in one coordinate direction at
the boundary. The accuracy obtained with piece-wise linear
basis functions, for example, is O(N-2 (logN)d-1)
with respect to the L2- and L-norm, if the
solution has bounded second mixed derivatives. This way,
the curse of dimensionality, i.e., the exponential
dependence O(Nd) of conventional approaches, is overcome
to some extent. For the energy norm, only O(N) degrees of
freedom are needed to give an accuracy of O(N-1). This
is why sparse grids are especially well-suited for problems
of very high dimensionality.
The sparse grid approach can be extended to nonsmooth
solutions by adaptive refinement methods. Furthermore, it
can be generalized from piecewise linear to higher-order
polynomials. Also, more sophisticated basis functions like
interpolets, prewavelets, or wavelets can be used in a
straightforward way.
We describe the basis features of sparse grids and report
the results of various numerical experiments for the
solution of elliptic PDEs as well as for other selected
problems such as numerical quadrature and data mining.